Solve the equation. $\dfrac{dy}{dx}=-\dfrac{1}{5xy}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\dfrac{C}{\sqrt[5]{|x|}}$ (Choice B) B $y=\pm\dfrac{1}{\sqrt[5]{|x|}}+C$ (Choice C) C $y=\pm \sqrt{-\dfrac{2\ln|x|}{5}+C}$ (Choice D) D $y=\pm \sqrt{-\dfrac{2\ln|x|}{5}}+C$
We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=-\dfrac{1}{5xy} \\\\ -5y\,dy&=\dfrac{1}{x}\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} -5y\,dy&=\dfrac{1}{x}\,dx \\\\ \int -5y\,dy&=\int \dfrac{1}{x}\,dx \\\\ -\dfrac{5y^2}{2}&=\ln|x|+C_1 \\\\ y^2&=-\dfrac{2\ln|x|}{5}+C \\\\ y&=\pm \sqrt{-\dfrac{2\ln|x|}{5}+C} \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\pm \sqrt{-\dfrac{2\ln|x|}{5}+C}$